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Properties

Label 2-1300-65.63-c1-0-6
Degree $2$
Conductor $1300$
Sign $0.863 - 0.503i$
Analytic cond. $10.3805$
Root an. cond. $3.22188$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.133 + 0.5i)3-s + (0.232 + 0.133i)7-s + (2.36 + 1.36i)9-s + (4.59 + 1.23i)11-s + (−3 − 2i)13-s + (2.86 − 0.767i)17-s + (−0.866 − 3.23i)19-s + (−0.0980 + 0.0980i)21-s + (0.133 + 0.0358i)23-s + (−2.09 + 2.09i)27-s + (−1.03 + 0.598i)29-s + (2.26 + 2.26i)31-s + (−1.23 + 2.13i)33-s + (3.23 − 1.86i)37-s + (1.40 − 1.23i)39-s + ⋯
L(s)  = 1  + (−0.0773 + 0.288i)3-s + (0.0877 + 0.0506i)7-s + (0.788 + 0.455i)9-s + (1.38 + 0.371i)11-s + (−0.832 − 0.554i)13-s + (0.695 − 0.186i)17-s + (−0.198 − 0.741i)19-s + (−0.0214 + 0.0214i)21-s + (0.0279 + 0.00748i)23-s + (−0.403 + 0.403i)27-s + (−0.192 + 0.111i)29-s + (0.407 + 0.407i)31-s + (−0.214 + 0.371i)33-s + (0.531 − 0.306i)37-s + (0.224 − 0.197i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.863 - 0.503i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 0.863 - 0.503i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.848194393\)
\(L(\frac12)\) \(\approx\) \(1.848194393\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 + (3 + 2i)T \)
good3 \( 1 + (0.133 - 0.5i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-0.232 - 0.133i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.59 - 1.23i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.86 + 0.767i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.866 + 3.23i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.133 - 0.0358i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.03 - 0.598i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.26 - 2.26i)T + 31iT^{2} \)
37 \( 1 + (-3.23 + 1.86i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.86 - 6.96i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-2.66 - 9.96i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + 7.46iT - 47T^{2} \)
53 \( 1 + (-8.46 - 8.46i)T + 53iT^{2} \)
59 \( 1 + (-13.7 + 3.69i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.23 - 10.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.86 + 0.767i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + 0.928T + 73T^{2} \)
79 \( 1 + 11.4iT - 79T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 + (0.794 - 2.96i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.23 - 2.13i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.808044639097059892857826715820, −9.070615746528879249853134214777, −8.048755298194196727655612639951, −7.21848121807806751865081437816, −6.56454312347139173860545133418, −5.35415682004052495437078582619, −4.61758241471623841922046260701, −3.75968386436507509998024796370, −2.49226641549696068668726998388, −1.17878983034632481734052860033, 1.00184314704626977858244282022, 2.10971480225005478917739279516, 3.67089760901643102967082369699, 4.22976954565868685737244794722, 5.48692138703710455907962805136, 6.42711715778246116329766043206, 7.03284428721703394449088201513, 7.87556543165886860623485617868, 8.857513314939863898013975193953, 9.621365815443754888267929038346

Graph of the $Z$-function along the critical line